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Theorem ofcc 33092
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcc.1 (𝜑𝐴𝑉)
ofcc.2 (𝜑𝐵𝑊)
ofcc.3 (𝜑𝐶𝑋)
Assertion
Ref Expression
ofcc (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofcc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofcc.2 . . . 4 (𝜑𝐵𝑊)
2 fnconstg 6776 . . . 4 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
31, 2syl 17 . . 3 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4 ofcc.1 . . 3 (𝜑𝐴𝑉)
5 ofcc.3 . . 3 (𝜑𝐶𝑋)
6 fvconst2g 7199 . . . 4 ((𝐵𝑊𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
71, 6sylan 580 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
83, 4, 5, 7ofcfval 33084 . 2 (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
9 fconstmpt 5736 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
108, 9eqtr4di 2790 1 (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {csn 4627  cmpt 5230   × cxp 5673   Fn wfn 6535  cfv 6540  (class class class)co 7405  f/c cofc 33081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-ofc 33082
This theorem is referenced by: (None)
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